The Meshless Local Petrov-Galerkin (MLPG) method has been employed to analyze the following linear and nonlinear solid mechanics problems: free and forced vibrations of a segmented bar and a cracked beam, pull-in instability of an electrostatically actuated microbeam, and plane strain deformations of incompressible hyperelastic materials. The Moving Least Squares (MLS) approximation is used to generate basis functions for the trial solution, and for the test functions. Local symmetric weak formulations are derived, and the displacement boundary conditions are enforced by the method of Lagrange multipliers. Three different techniques are employed to enforce continuity conditions at the material interfaces: Lagrange multipliers, jump functions, and MLS basis functions with discontinuous derivatives. For the electromechanical problem, the pull-in voltage and the corresponding deflection are extracted by combining the MLPG method with the displacement iteration pull-in extraction algorithm. The analysis of large deformations of incompressible hyperelastic materials is performed by using a mixed pressure-displacement formulation. For every problem studied, computed results are found to compare well with those obtained either analytically or by the Finite Element Method (FEM). For the same accuracy, the MLPG method requires fewer nodes but more CPU time than the FEM. / Ph. D.

Plane strain deformations of an elastoplastic material are studied using numerical methods. In the first chapter, a meshless formulation of the static small strain elastic-plastic problem is formulated using the meshless local Petrov-Galerkin method. The code is validated against the small strain plasticity routines in the commercial finite element code ABAQUS for two basic configurations with loading, unloading, and reloading. The results are found to agree within 5%. The validated code is then used to analyze the stress intensity factor (SIF) in a double edge-cracked plate. Deformations of the plate are studied both with and without exploiting the symmetry conditions. The penalty method is used to enforce the essential boundary condition in the former case. When analyzing the deformations of the entire plate, the diffraction method is employed in order to introduce the discontinuity in the displacement field across the crack faces. The log-log and a higher order extrapolation technique due to Dally and Berger (1996) are used to calculate the SIF. It is found that the penalty method was inadequate to enforce the essential boundary conditions in the vicinity of the crack tip and that in this region the deformations were oscillatory. Consequently, the SIF calculation using the higher order technique was not accurate. It is also found that for a small plastic zone (3% of the cracked length) the SIFs do not differ significantly from their values for the corresponding linear elastic problem. In the second chapter, a finite element formulation of the dynamic deformations of a micro-porous thermoviscoplastic solid is formulated. The heat conduction in a material is assumed to be governed by a hyperbolic heat equation; thus thermal and mechanical waves propagate with finite speeds. The formation and propagation of an adiabatic shear band (ASB) inplane strain tensile deformations is studied for eleven materials. The ASB is assumed to form when the maximum shear stress has been reduced to 80% of its peak value at a point and it is deforming plastically. The materials are ranked according their susceptibility to the formation of an ASB. A parametric study of the effect of the initial defect strength where the defect is assumed through an initially inhomogeneous distribution of porosity, the thermal conductivity, the thermal wave speed, and the applied strain-rate upon the ASB initiation and propagation is conducted. It is found that the susceptibility ranking for this configuration differs somewhat from that previously found for simple shear and torsion of thin-walled tubes. It is also found that thermal conductivity influences ASB initiation and propagation only for materials with large values of · and that for such materials an adiabatic model may not be adequate. The effects of initial defect strength and the nominal strain-rates are both found to be consistent with simple shearing studies except that the ASB propagation speed was found to decrease with increasing nominal strain-rate. It is found that the criterion employed for ASB initiation accurately predicts the onset of the collapse of the total axial load applied to the body. In the final chapter, the formulation from the previous chapter is modified to permit the formation and propagation of brittle and ductile fracture. Deformations of the impact loaded double edge-crack specimen of Kalthoff and Winkler (1987) are studied. The brittle to ductile failure mode transition with increasing impact speed was found. Previous studies have focused on identifying the transition speed and did not allow for crack propagation. In this study, crack propagation is achieved through a nodal release algorithm and interpenetration of the crack surfaces is prevented using stiff-spring contact elements. Brittle fracture is assumed to occur when the maximum tensile principal stress achieves a critical value and the ductile fracture is assumed to occur when the effective plastic strain reaches a critical value. It is found that the transition speed for 4340 steel is approximately 54 m/s. For the brittle failure, the stress field is found to be significantly modified by the propagating crack and in the vicinity of the propagating crack the field is mode-I dominant. The crack formed through brittle fracture is found to completely propagate through the plate. For the ductile failure, the distribution of effective plastic strain about the crack tip is not significantly altered by the formation of the crack. The temperature rise in the vicinity of the ductile crack is found to be approximately 45% of the melting temperature of the material. / Ph. D.

The meshless local Petrov-Galerkin (MLPG) method is used to numerically find an approximate solution of plane strain/stress linear elastostatic and elastodynamic problems. The MLPG method requires only a set of nodes both for the interpolation of the solution variables and the evaluation of various integrals appearing in the problem formulation. The monomial basis functions in the MLPG formulation have been enriched with those for the linear elastic fracture mechanics solutions near a crack tip. Also, the diffraction and the visibility criteria have been added to make the displacement field discontinuous across a crack. A computer code has been developed in Fortran and validated by comparing computed solutions of three static and one dynamic problem with their analytical solutions. The capabilities of the code have been extended to analyze contact problems in which a displacement component and the complementary traction component are prescribed at the same point of the boundary. The code has been used to analyze stress and deformation fields near a crack tip and to find the stress intensity factors by using contour integrals, the equivalent domain integrals and the J-integral and from the intercepts with the ordinate of the plots, on a logarithmic scale, of the stress components versus the distance ahead of the crack tip. We have also computed time histories of the stress intensity factors at the tips of a central crack in a rectangular plate with plate edges parallel to the crack loaded in tension. These are found to compare favorably with those available in the literature. The code has been used to compute time histories of the stress intensity factors in a double edge-notched plate with the smooth edge between the notches loaded in compression. It is found that the deformation fields near the notch tip are mode-II dominant. The mode mixity parameter can be changed in an orthotropic plate by adjusting the ratio of the Young's moduli in the axial and the transverse direction. The plane strain problem of compressing a linear elastic material confined in a rectangular cavity with rough horizontal walls and a smooth vertical wall has been studied with the developed code. Computed displacements and stresses are found to agree well with the analytical solution of the problem obtained by the Laplace transform technique. The Appendix describes the analysis with the finite element code ABAQUS of the dependence of the energy release rate upon the crack length in a polymeric disk enclosed in a steel ring and having a star shaped hole at its center. A starter crack is assumed to exist in one of the leaflets of the hole. The disk is loaded either by a pressure acting on the surfaces of the hole and the crack or by a temperature rise. Computed values of the energy release rate obtained by modeling the disk material as Hookean are found to be about 30% higher than those obtained when the disk material is modeled as Mooney-Rivlin. The latter set of results accounts for both material and geometric nonlinearities. / Ph. D.